Poisson brackets

Poisson brackets are very popular in theoretical physics. In the case of a classical mechanical system described in terms of a Hamiltonian the underlying mathematical structure is a finite-dimensional manifold $M$ and a symplectic structure $\omega$. This object has an inverse $\omega^{-1}$. The Poisson bracket of two functions $f$ and $g$ on $M$ is defined in terms of their exterior derivatives as $\omega^{-1}(df,dg)$. There is no problem here for the mathematician interested in understanding this. What is more difficult is the definition of the Poisson bracket for field theories. Formally this corresponds to allowing the manifold $M$ to be infinite dimensional. Consider for instance a scalar field $\phi$, in one space dimension for maximum simplicity. Thus we are considering functions $\phi (t,\theta)$ and I suppose (simplicity again) that they are periodic in $\theta$. The Lagrangian density is $\frac12 (\phi_t^2-\phi_\theta^2)$. The momentum $\pi$ conjugate to $\phi$ is defined to be the time derivative $\phi_t$. In this case the phase space $M$ is formally the space of functions $(\phi,\pi)$.I will not try to specify what kind of functions – for example we can think of them as being smooth.Given two functionals $F$ and $G$ (i.e. two functions on the infinite dimensional space $M$) the Poisson bracket is defined by the formula $\int_{S^1} \frac{\delta F}{\delta\phi}\frac{\delta G}{\delta\pi}-\frac{\delta F}{\delta\pi}\frac{\delta G}{\delta\phi}$. The derivatives here are functional derivatives. Now, feeling certain that my physicist colleagues will react with amusement or incomprehension, I must admit that I do not understand what a functional derivative is. This formula is a hieroglyph for me.

Let us not be discouraged. The functional derivative looks something like a variational derivative, a concept which is more familiar to me. This basically just means that if I have a functional which is the integral over $S^1$ of a function of $\phi$ and $\pi$ then it is possible to consider variations $(\phi(\lambda),\pi(\lambda))$, intuitively curves in the manifold $M$, and differentiate them with respect to the parameter $\lambda$. If the space $M$ can be defined as a decent infinite-dimensional manifold (e.g. a Banach manifold) then this can be related to constructions of differential geometry on that manifold. The variational derivative just corresponds to the exterior derivative and is a one-form. If we start with smooth functions then this object takes values in the dual space. In other words we encounter distributions. Now in the presence of a volume form many distributions can be identified with functions. The one-form is then integration against that function, which could be called the functional derivative and used in the definition of the Poisson bracket.If the functionals are integrals of expressions depending pointwise on $\phi$ and $\pi$ then this is nothing other than the partial derivative with respect to the same quantity. Notice that with this definition we can reproduce the fact that the momentum $\pi$ is the functional derivative of the Lagrangian with respect to $\phi_t$ If the functional is allowed to depend on the spatial derivative of $\phi$ then a further complication is added since in that case it is necessary to integrate by parts in space to compute the function corresponding to the distribution.If we can interpret the functional derivatives as functions of $\theta$ then the Poisson bracket can be written as $\int_{S^1}\frac{\delta F}{\delta\phi}(\theta)\frac{\delta G}{\delta\pi}(\theta)-\frac{\delta F}{\delta\pi}(\theta)\frac{\delta G}{\delta\phi}(\theta)d\theta$.

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